If `A=[{:(2,1),(0,5):}]`, find `|A^(-1)|`

Answer

To find the determinant of the inverse of the matrix \( A = \begin{pmatrix} 2 & 1 \\ 0 & 5 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated using the formula:
\[ |A| = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = 1 \)
- \( c = 0 \)
- \( d = 5 \)

Substituting these values into the formula:
\[ |A| = (2)(5) - (1)(0) = 10 - 0 = 10 \]

### Step 2: Check if A is Invertible
A matrix is invertible if its determinant is non-zero. Since \( |A| = 10 \), which is non-zero, matrix \( A \) is invertible.

### Step 3: Use the Property of Determinants
The determinant of the inverse of a matrix is given by the formula:
\[ |A^{-1}| = \frac{1}{|A|} \]
Using the determinant we calculated in Step 1:
\[ |A^{-1}| = \frac{1}{10} \]

### Final Answer
Thus, the determinant of the inverse of matrix \( A \) is:
\[ |A^{-1}| = \frac{1}{10} \]

---